Hi, just saw your message. I'm Chinese too, and I'm in my 30's. My prior degrees are not in classics, because I needed a job to secure my visa and food. Glad to see people from my home country having similar interests!
I fully understand your concern for age, as it matters in some industries and in some places. But I think it doesn't matter that much here in the US, in academia (maybe in pure mathematics it matters, haha). The posts above gave sufficient evidence. I can give you my own evidence. I started learning the ancient languages not long ago in one of my local colleges, and I got PhD admission offers this year. I didn't have the two years of latin/Greek each as you are now. I spent my MA (not in classics) figuring out what I want to do and then did the language classes / seminars as much as possible.
Well, you certainly can choose to save for the future and do something in your 50's or later. But life is unpredictable. For me, I simply want to make sure, if I got hit by a car by accident and went to another unknown realm, I would not regret at the moment I left this world. I would suggest taking some time to think about what you want for your life and how much you want it, and then just start trying.
Once you are determined, you can sort out other constraints like finance. As posts above say, there's funded MAs. I do think MAs are better than post-Bac, as it will give you sufficient time to explore, and at the same time you can find some depth.
Good luck with your future planning and studying!